In the Black Scholes setting, here is how my understanding is of how we derive the PDE for the value of an option.

  1. We assume that the price of the option is Markovian in our state variable $S_t$. Then we can use Ito's formula to get the SDE of our option process.
  2. We form a portfolio of the underlying and the option. Using Ito, we can know get a SDE for our portfolio process.

  3. Using the SDE of the portfolio we make it independent of $dS_t$ (and thus also $dW_t$, i.e. it is locally riskless) by choosing our portfolio weights appropiately.

  4. Using arbitrage arguments, we know then that the remaining $dt$-term in the portfolio process must equal $r(t)$.

  5. This condition is the Black Scholes PDE.
  6. Using a Feynman-Kac representation, we can convert this PDE representation to a discounted expectation but under a "different" (risk-neutral) measure.

Now, for a non-Black Scholes setting (e.g. stochastic volatility), how must this procedure be modified to produce the PDE for the option AND the risk-neutral expectation? In particular, I have 2 questions:

  1. Do we still assume that the price of the option is Markovian in $S_t$ but now also in $\sigma_t$, the volatility?
  2. When forming the portfolio, I need to ensure that the $dS_t$ and the $d\sigma_t$ terms in the SDE of the portfolio process disappear, so that we have a locally riskless portfolio. But how can I do this? If I am investing in the option + the underlying, I have 2 variables but 3 equations.

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